Question: What is the slope of the line tangent to $f(x) = x^{2}+x-7$ at $x = -2$ ?
Solution: The slope of the tangent line is $ \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ $ = \lim_{h \to 0} \frac{((x+h)^{2}+x+h-7) - (x^{2}+x-7)}{h}$ $ = \lim_{h \to 0} \frac{(x^{2}+2x h+h^{2}+x+h-7) - (x^{2}+x-7)}{h}$ $ = \lim_{h \to 0} \frac{x^{2}+2(x h)+h^{2}+x+h-7-x^{2}-x+7}{h}$ $ = \lim_{h \to 0} \frac{2(x h)+h^{2}+h}{h}$ $ = \lim_{h \to 0} 2x+h+1$ $ = 2x+1$ $ = (2)(-2)+1$ $ = -3$